Permutation Polynomials and Orthomorphism Polynomials of Degree Six

A classic paper of Dickson claims to give a complete list of permutation polynomials of degree less than 6 over arbitrary finite fields, and degree 6 over finite fields of odd characteristic. However, recent published statements have suggested that Dickson's classification was incomplete in the degree 6 case. We uncover the reason for this confusion, and establish the true list of degree 6 permutation polynomials over all finite fields. As an application, we determine the complete list of degree 6 orthomorphism polynomials. Additionally, we note that one of the derived families of permutation polynomials provides a counterexample to a published conjecture of Mullen.